Добавил:

Ravochking Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Кузбасский государственный технический университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

_eBook__-_Math_Calculus_Bible.pdf

Скачиваний:

Добавлен:

23.08.2019

Размер:

1.23 Mб

Скачать

☆

<<< < Предыдущая 13 14 15 16 17 18 19 20 21 22 23 24 2526 / 3726 27 28 29 30 31 32 33 34 35 36 37 > Следующая >>>

280

CHAPTER 6. TECHNIQUES OF INTEGRATION

65.

sin mx sin nx dx =

−n

−

m + n

+ C; m2 6= n2

−

sin(m + n)x

sin(m

n)x

66.

cos mx cos nx dx =

−

+ C; m2 6= n2

m + n

−

sin(m + n)x

sin(m

n)x

6.4Trigonometric Integrals

The trigonometric integrals are of two types. The integrand of the ﬁrst type consists of a product of powers of trigonometric functions of x. The integrand of the second type consists of sin(nx) cos(mx), sin(nx) sin(mx) or cos(nx) cos(mx). By expressing all trigonometric functions in terms of sine and cosine, many trigonometric integrals can be computed by using the following theorem.

Theorem 6.4.1 Suppose that m and n are integers, positive, negative, or zero. Then the following reduction formulas are valid:

1.sinn x

2.sinn−2


	n				n		Z
dx =	−1		sinn−1 x cos x +		(n − 1)

1				sinn−1 x cos x +		n
x dx =
		n − 1				n − 1

sinn−2 x dx, n > 0

sinn x dx, n ≤ 0

3.(sin x)−1 dx = csc x dx = ln | csc x−cot x|+c or − ln | csc x+cot x|+c

4.cosn x dx = n1

5.cosn−2 x dx =

6.(cos x)−1 dx =

			n	Z
cosn−1 x sin x +			n − 1	cosn−2 x dx, n > 0
cosn−1 x sin x +				cosn−2 x dx, n > 0
n − 1				n − 1	Z	≤
−1		cosn−1 x sin x +		n	cosn x dx, n		0
		cosn−1 x sin x +			cosn x dx, n		0
Z	sec x dx = ln \| sec x + tan x\| + c

7.sinnx cos2m+1 x dx = R sinn x(1 − sin2 x)m cos x dx

=R un(1 − u2)mdu, u = sin x, du = cos x dx

6.4. TRIGONOMETRIC INTEGRALS

281

8.sin2n+1x cosm x dx = R cosm x(1 − cos2 x)n sin x dx

= − R um(1 − u2)ndu, u = cos x, du = − sin x dx

9.sin2n x cos2m x dx = R (1 − cos2 x)n cos2m x dx

=R (1 − sin2 x)m sin2n x dx

m − n

10.

sin(nx) cos(mx)dx =

−1

cos(m + n)x

cos(m − n)x

+ c, m2 = n2

11.

sin(mx) sin(mx) dx =

−

+ c, m2 6= n2

m + n

−

sin(m + n)x

sin(m

n)x

12.

cos(mx) cos(mx) dx =

−

+ c, m2 6= n2

m + n

−

sin(m + n)x

sin(m

n)x

Corollary. The following integration formulas are valid:

tann

1 u

− Z

13.

tann u du =

−

tann−2 u d

n − 1

14.

secn u du =

secn−2 x tan x +

− 2

secn−2 x dx

n − 1

15.

cscn u du =

−1

cscn−2 x cot x +

− 2

cscn−2 x dx

Exercises 6.4 Evaluate each of the following integrals.

1.	Z	sin5 x dx	2.	Z	cos4 x dx
3.	Z	tan5 x dx	4.	Z	cot4 x dx
5.	Z	sec5 x dx	6.	Z	csc4 x dx

282			CHAPTER 6. TECHNIQUES OF INTEGRATION
7.	Z	sin5 x cos4 x dx	8.	Z	sin3 x cos5 x dx
9.	Z	sin4 x cos3 x dx	10.	Z	sin2 x cos4 x dx
11.	Z	tan5 x sec4 x dx	12.	Z	cot5 x csc4 x dx
13.	Z	tan4 x sec5 x dx	14.	Z	cot4 x csc5 x dx
15.	Z	tan4 x sec4 x dx	16.	Z	cot4 x csc4 x dx
17.	Z	tan3 x sec3 x dx	18.	Z	cot3 x csc3 x dx
19.	Z	sin 2x cos 3x dx	20.	Z	sin 4x cos 4x dx
21.	Z	sin 3x cos 3x dx	22.	Z	sin 2x sin 3x dx
23.	Z	sin 4x sin 6x dx	24.	Z	sin 3x sin 5x dx
25.	Z	cos 3x cos 5x dx	26.	Z	cos 2x cos 4x dx
27.	Z	cos 3x cos 4x dx	28.	Z	sin 4x cos 4x dx

6.5Trigonometric Substitutions

Theorem 6.5.1 (a2 − u2 Forms). Suppose that u = a sin t, a > 0. Then

6.5. TRIGONOMETRIC SUBSTITUTIONS

283

a2 − u2 = a2 cos2 t,

√

= a cos t, t = arcsin(u/a),

du = a cos tdt,

a2 − u2

√

− u

sin t =

, cos t =

, tan t =

√a2 − u2

cot t =

√

a2 − u2

sec t =

csc t

√a2 − u2

graph

The following integration formulas are valid:

a2 − u2 = −2 ln |a2 − u2| + c

udu

a2 − u2

a + u

1 ln

a − u

+ c =

arctanh

+ c

udu

√

= −√a2 − u2 + c

a2 − u2

√

= arcsin

+ c

a2 − u2

√

−

+ c

u√a2 − u2 = a

ln u

u−

Z √

du =

arcsin a

+ 2 u√

a2 − u2

− u2 + c

Proof. The proof of this theorem is left as an exercise.

284

CHAPTER 6.

TECHNIQUES OF INTEGRATION

Theorem 6.5.2 (a2 + u2 Forms). Suppose that u = a tan t, a > 0. Then

2 tdt, a2

a2 sec2 t, √

= a sec t, t = arctan

a2 + u2

secu

sin t =

√

cos t =

√

tan t =

a2 + u2

√

+ u2

a2 + u2

csc t =

sec t =

cot t =

graph

Proof. The proof of this theorem is left as an exercise.

The following integration formulas are valid:

udu

+ u2 + c

a2 + u2

arctan

+ c

a2 + u2

udu

√

= √a2 + u2 + c

a2 + u2

√a2 + u2

= ln

u + √a2 + u2

+ c

√

a2 + u2

+ c

u√a2 + u2 = a ln

− u

√

du =

+ c

a + u

2 u a + u

ln u + a + u

Theorem 6.5.3 (u

− a

Forms) Suppose that u = a sec t, a > 0. Then

a2 = a2 tan2 t,

√

a sec t tan t dt,

−

= a tan t, t = arcsec

√

sin t =

− a

, cos t =

tan t =

− a

√

csc t =

, sec t =

cot t =

u2 − a2

6.5. TRIGONOMETRIC SUBSTITUTIONS

285

graph

Proof. The proof of this theorem is left as an exercise.

The following integration formulas are valid:

− a2 =

2 ln u2 − a2 + c

udu

−

u + a

− a

+ c

√

udu

√

= u

− a + c

u2 − a2

√u2

− a2

= ln u + √u2 − a2 + c

u√u2 − a2

= a

arcsec

+ c

Z √u2 − a2 du =

u√u2 − a2

−

ln u + √u2

Exercises 6.5 Prove each of the following formulas:

u du

ln |a2 − u2| + C

−

a2 − u2

a + u

a − u

+ C

√

u du

√

− u

= − a

+ C

a2 − u2

√

= arcsin

+ C, a > 0

a2 − u2

u√a2

u2 = a

−

+ C

u−

√a2

−

− a2 + c

286

CHAPTER 6.

TECHNIQUES OF INTEGRATION

Z √a2 − u2

du =

arcsin

u√a2 − u2 + C, a > 0

a2 + u2 = 2 ln a2

+ u2 + C

u du

arctan

+ C

a2 + u2

√a2 + u2

= √a2 + u2 + C

u du

10.

√a2 + u2

= ln u + √a2 + u2

+ C

− u + C

11.

u√a2 + u2

= a ln

√a

+ u

12.

Z √a2 + u2

du =

+ u2

√a2 + u2

+ C

2 u√a2

ln u +

13.

u du

− a2

+ C

u2 − a2

u + a

14.

− a

+ C

u du

√

= √u

15.

− a + C

u2 − a2

+ C

16.

√u2 − a2

= ln u + √u2 − a2

17.

u√

arcsec

+ C

u2 − a2

18.

Z √u2 − a2

du =

u√u2

− a2

−

u +

√u2 − a2

+ C

Evaluate each of the following integrals:

6.5.

TRIGONOMETRIC SUBSTITUTIONS

19.

√4 − x2

20.

√4 − x2

x dx

22.

4 − x2

23.

9 + x2

x dx

25.

√9 + x2

26.

√9 + x2

x dx

28.

x2 − 16

29.

√x2 − 16

x dx

31.

x√x2

− 4

32.

x√9 − x2

Z √

34.

9 − x2

35.

4 − 9x2

37.

√

38.

√

4 + x2

(9 − x2)2

−

40.

41.

(x2 − 16)2

√

+ x2

43.

44.

−

46.

x2√4 − x2

47.

x2√x2

− 4

49.

x2 − 4x + 12

50.

√4x − x2

52.

4x − x2

53.

√x2 − 2x + 5

55.

√x2 x

2x + 5

56.

x2 + 4x + 13 dx

−

287

21.

4x− x2

24.

9 + x2

27.

x2 − 16

x dx

30.

√

x2 16

−

33.

x√

x2 + 16

36.

√

1 x2

−

39.

(9 + x2)2

42.

(4 + x2)3/2

45.

x2√x2 + 4

48.

x2 − 2x + 5

51.

√x2 − 4x + 12

54.

x2 − 4x − 12

57.(5 − 4x − x2)1/2dx

<<< < Предыдущая 13 14 15 16 17 18 19 20 21 22 23 24 2526 / 3726 27 28 29 30 31 32 33 34 35 36 37 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
19.04.20201.71 Mб1zotova_z_m_politicheskie_partii_rossii_organizatsiya_i_deyat.pdf
#
03.10.2020561.53 Кб2zueva_ov_suraikina_ea_turistskie_resursy_i_infrastruktura_sa.pdf
#
18.06.2020836.8 Кб68zueva_ta_ivanova_en_stilisticheskaia_pravka_i_redaktirovanie.pdf
#
23.08.201911.43 Mб16[ebook4everything.com]Steve Jobs - Isaacson_ Walter.pdf
#
19.11.2019775.76 Кб1_brosh_n1_sait.pdf
#
23.08.20191.23 Mб8_eBook__-_Math_Calculus_Bible.pdf
#
22.08.20195 Mб24_Андрей Парабеллум, Личная власть. 7 отважных шагов.pdf
#
22.08.20192.72 Mб6_Андрей Парабеллум, Развитие бизнеса.pdf
#
22.08.2019624.47 Кб11_Бизнес без правил.pdf
#
22.08.20193.68 Mб7_Бразговский А.С., 78 бюджетных методов увеличить продажи.pdf
#
23.08.20197.72 Mб12_Васильев В.В., История философии.pdf